A coreflective subcategory is a full subcategory whose inclusion functor has a right adjoint $R$ (a cofree functor):
The dual concept is that of a reflective subcategory. See there for more details.
Vopěnka's principle is equivalent to the statement:
For $C$ a locally presentable category, every full subcategory $D \hookrightarrow C$ which is closed under colimits is a coreflective subcategory.
This is (AdamekRosicky, theorem 6.28).
the inclusion of Kelley space?s into Top, where the right adjoint “kelleyfies”
the inclusion of torsion abelian groups into Ab, where the right adjoint takes the torsion subgroup.
the inclusion of groups into monoids, where the right adjoint takes a monoid to its group of units.